Abstract

Quantum backflow is a counterintuitive phenomenon in which a forward-propagating quantum particle propagates locally backwards. The actual counter-propagation property associated with this delicate interference phenomenon has not been observed to date in any field of physics, to the best of our knowledge. Here, we report the observation of an analog optical effect, namely, transverse optical backflow where a beam of light propagating to a specific transverse direction is measured locally to propagate in the opposite direction. This observation is relevant to any physical system supporting coherent waves.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  28. R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Superoscillating electron wave functions with subdiffraction spots,” Phys. Rev. A 95, 031802 (2017).
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  29. R. Remez and A. Arie, “Super-narrow frequency conversion,” Optica 2, 472–475 (2015).
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  30. Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the temporal resolution limit by superoscillating optical beats,” Phys. Rev. Lett. 119, 043903 (2017).
    [Crossref]
  31. Y. Eliezer, B. K. Singh, L. Hareli, A. Arie, and A. Bahabad, “Experimental realization of structured super-oscillatory pulses,” Opt. Express 26, 4933–4941 (2018).
    [Crossref]
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    [Crossref]
  33. M. Palmero, E. Torrontegui, J. Muga, and M. Modugno, “Detecting quantum backflow by the density of a Bose-Einstein condensate,” Phys. Rev. A 87, 053618 (2013).
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    [Crossref]
  35. G. Yuan, E. T. Rogers, and N. I. Zheludev, ““Plasmonics” in free space: observation of giant wavevectors, vortices, and energy backflow in superoscillatory optical fields,” Light: Sci. Appl. 8, 2 (2019).
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    [Crossref]
  37. E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38, 3546–3549 (2013).
    [Crossref]
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    [Crossref]
  40. J. Tollaksen, “Novel relationships between superoscillations, weak values, and modular variables,” J. Phys. Conf. Ser. 70, 012016 (2007).
    [Crossref]
  41. M. Berry and P. Shukla, “Pointer supershifts and superoscillations in weak measurements,” J. Phys. A 45, 015301 (2011).
    [Crossref]
  42. Y. Aharonov, S. Popescu, and J. Tollaksen, “A time symmetric formulation of quantum mechanics,” Phys. Today 63 (11), 27 (2010).
    [Crossref]
  43. M. Berry, “Five momenta,” Eur. J. Phys. 34, 1337 (2013).
    [Crossref]
  44. V. C. Coffey, “Advances in standoff detection make the world safer,” Photon. Spectra 47, 44–47 (2013).
  45. J. L. Gottfried, F. C. De Lucia, C. A. Munson, and A. W. Miziolek, “Laser-induced breakdown spectroscopy for detection of explosives residues: a review of recent advances, challenges, and future prospects,” Anal. Bioanal. Chem. 395, 283–300 (2009).
    [Crossref]

2019 (1)

G. Yuan, E. T. Rogers, and N. I. Zheludev, ““Plasmonics” in free space: observation of giant wavevectors, vortices, and energy backflow in superoscillatory optical fields,” Light: Sci. Appl. 8, 2 (2019).
[Crossref]

2018 (1)

2017 (6)

Y. Eliezer and A. Bahabad, “Super defocusing of light by optical sub-oscillations,” Optica 4, 440–446 (2017).
[Crossref]

T. Zacharias, B. Hadad, A. Bahabad, and Y. Eliezer, “Axial sub-Fourier focusing of an optical beam,” Opt. Lett. 42, 3205–3208 (2017).
[Crossref]

B. K. Singh, H. Nagar, Y. Roichman, and A. Arie, “Particle manipulation beyond the diffraction limit using structured super-oscillating light beams,” Light: Sci. Appl. 6, e17050 (2017).
[Crossref]

R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Superoscillating electron wave functions with subdiffraction spots,” Phys. Rev. A 95, 031802 (2017).
[Crossref]

Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the temporal resolution limit by superoscillating optical beats,” Phys. Rev. Lett. 119, 043903 (2017).
[Crossref]

H. Bostelmann, D. Cadamuro, and G. Lechner, “Quantum backflow and scattering,” Phys. Rev. A 96, 012112 (2017).
[Crossref]

2016 (2)

Y. Eliezer and A. Bahabad, “Super-oscillating airy pattern,” ACS Photon. 3, 1053–1059 (2016).
[Crossref]

F. Albarelli, T. Guaita, and M. G. Paris, “Quantum backflow effect and nonclassicality,” Int. J. Quantum Inf. 14, 1650032 (2016).
[Crossref]

2015 (2)

2013 (8)

A. M. Wong and G. V. Eleftheriades, “An optical super-microscope for far-field, real-time imaging beyond the diffraction limit,” Sci. Rep. 3, 1715 (2013).
[Crossref]

E. Greenfield, R. Schley, I. Hurwitz, J. Nemirovsky, K. G. Makris, and M. Segev, “Experimental generation of arbitrarily shaped diffractionless superoscillatory optical beams,” Opt. Express 21, 13425–13435 (2013).
[Crossref]

M. Palmero, E. Torrontegui, J. Muga, and M. Modugno, “Detecting quantum backflow by the density of a Bose-Einstein condensate,” Phys. Rev. A 87, 053618 (2013).
[Crossref]

E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38, 3546–3549 (2013).
[Crossref]

B. Tamir and E. Cohen, “Introduction to weak measurements and weak values,” Quanta 2, 7–17 (2013).
[Crossref]

E. T. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15, 094008 (2013).
[Crossref]

M. Berry, “Five momenta,” Eur. J. Phys. 34, 1337 (2013).
[Crossref]

V. C. Coffey, “Advances in standoff detection make the world safer,” Photon. Spectra 47, 44–47 (2013).

2011 (1)

M. Berry and P. Shukla, “Pointer supershifts and superoscillations in weak measurements,” J. Phys. A 45, 015301 (2011).
[Crossref]

2010 (2)

Y. Aharonov, S. Popescu, and J. Tollaksen, “A time symmetric formulation of quantum mechanics,” Phys. Today 63 (11), 27 (2010).
[Crossref]

M. Berry, “Quantum backflow, negative kinetic energy, and optical retro-propagation,” J. Phys. A 43, 415302 (2010).
[Crossref]

2009 (2)

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
[Crossref]

J. L. Gottfried, F. C. De Lucia, C. A. Munson, and A. W. Miziolek, “Laser-induced breakdown spectroscopy for detection of explosives residues: a review of recent advances, challenges, and future prospects,” Anal. Bioanal. Chem. 395, 283–300 (2009).
[Crossref]

2007 (3)

J. Tollaksen, “Novel relationships between superoscillations, weak values, and modular variables,” J. Phys. Conf. Ser. 70, 012016 (2007).
[Crossref]

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical super-resolution through super-oscillations,” J. Opt. A 9, S285 (2007).
[Crossref]

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007).
[Crossref]

2006 (1)

M. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A 39, 6965 (2006).
[Crossref]

2005 (1)

M. Penz, G. Grübl, S. Kreidl, and P. Wagner, “A new approach to quantum backflow,” J. Phys. A 39, 423 (2005).
[Crossref]

2002 (1)

J. Damborenea, I. Egusquiza, G. Hegerfeldt, and J. Muga, “Measurement-based approach to quantum arrival times,” Phys. Rev. A 66, 052104 (2002).
[Crossref]

2000 (1)

J. G. Muga and C. R. Leavens, “Arrival time in quantum mechanics,” Phys. Rep. 338, 353–438 (2000).
[Crossref]

1999 (1)

J. Muga, J. Palao, and C. Leavens, “Arrival time distributions and perfect absorption in classical and quantum mechanics,” Phys. Lett. A 253, 21–27 (1999).
[Crossref]

1994 (1)

A. Bracken and G. Melloy, “Probability backflow and a new dimensionless quantum number,” J. Phys. A 27, 2197 (1994).
[Crossref]

1988 (1)

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett 60, 1351 (1988).
[Crossref]

1969 (3)

G. R. Allcock, “The time of arrival in quantum mechanics I. Formal considerations,” Ann. Phys. (N. Y.) 53, 253–285 (1969).
[Crossref]

G. R. Allcock, “The time of arrival in quantum mechanics II. The individual measurement,” Ann. Phys. (N. Y.) 53, 286–310 (1969).
[Crossref]

G. Allcock, “The time of arrival in quantum mechanics III. The measurement ensemble,” Ann. Phys. (N. Y.) 53, 311–348 (1969).
[Crossref]

1961 (1)

D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[Crossref]

1952 (1)

G. T. Di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento 9, 426–438 (1952).
[Crossref]

1943 (1)

S. A. Schelkunoff, “A mathematical theory of linear arrays,” Bell Syst. Tech. J. 22, 80–107 (1943).
[Crossref]

Aharonov, Y.

Y. Aharonov, S. Popescu, and J. Tollaksen, “A time symmetric formulation of quantum mechanics,” Phys. Today 63 (11), 27 (2010).
[Crossref]

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett 60, 1351 (1988).
[Crossref]

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, The Mathematics of Superoscillations (American Mathematical Society, 2017), Vol. 247.

Albarelli, F.

F. Albarelli, T. Guaita, and M. G. Paris, “Quantum backflow effect and nonclassicality,” Int. J. Quantum Inf. 14, 1650032 (2016).
[Crossref]

Albert, D. Z.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett 60, 1351 (1988).
[Crossref]

Allcock, G.

G. Allcock, “The time of arrival in quantum mechanics III. The measurement ensemble,” Ann. Phys. (N. Y.) 53, 311–348 (1969).
[Crossref]

Allcock, G. R.

G. R. Allcock, “The time of arrival in quantum mechanics II. The individual measurement,” Ann. Phys. (N. Y.) 53, 286–310 (1969).
[Crossref]

G. R. Allcock, “The time of arrival in quantum mechanics I. Formal considerations,” Ann. Phys. (N. Y.) 53, 253–285 (1969).
[Crossref]

Arie, A.

Y. Eliezer, B. K. Singh, L. Hareli, A. Arie, and A. Bahabad, “Experimental realization of structured super-oscillatory pulses,” Opt. Express 26, 4933–4941 (2018).
[Crossref]

B. K. Singh, H. Nagar, Y. Roichman, and A. Arie, “Particle manipulation beyond the diffraction limit using structured super-oscillating light beams,” Light: Sci. Appl. 6, e17050 (2017).
[Crossref]

R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Superoscillating electron wave functions with subdiffraction spots,” Phys. Rev. A 95, 031802 (2017).
[Crossref]

R. Remez and A. Arie, “Super-narrow frequency conversion,” Optica 2, 472–475 (2015).
[Crossref]

Bahabad, A.

Bartal, G.

Bent, N.

Berry, M.

M. Berry, “Five momenta,” Eur. J. Phys. 34, 1337 (2013).
[Crossref]

M. Berry and P. Shukla, “Pointer supershifts and superoscillations in weak measurements,” J. Phys. A 45, 015301 (2011).
[Crossref]

M. Berry, “Quantum backflow, negative kinetic energy, and optical retro-propagation,” J. Phys. A 43, 415302 (2010).
[Crossref]

M. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A 39, 6965 (2006).
[Crossref]

M. Berry, Quantum Coherence and Reality: In Celebration of the 60th Birthday of Yakir Aharonov (World Scientific, 1994).

Blau, Y.

Bolduc, E.

Bostelmann, H.

H. Bostelmann, D. Cadamuro, and G. Lechner, “Quantum backflow and scattering,” Phys. Rev. A 96, 012112 (2017).
[Crossref]

Boyd, R. W.

Bracken, A.

A. Bracken and G. Melloy, “Probability backflow and a new dimensionless quantum number,” J. Phys. A 27, 2197 (1994).
[Crossref]

Cadamuro, D.

H. Bostelmann, D. Cadamuro, and G. Lechner, “Quantum backflow and scattering,” Phys. Rev. A 96, 012112 (2017).
[Crossref]

Chen, Y.

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007).
[Crossref]

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical super-resolution through super-oscillations,” J. Opt. A 9, S285 (2007).
[Crossref]

Coffey, V. C.

V. C. Coffey, “Advances in standoff detection make the world safer,” Photon. Spectra 47, 44–47 (2013).

Cohen, E.

B. Tamir and E. Cohen, “Introduction to weak measurements and weak values,” Quanta 2, 7–17 (2013).
[Crossref]

Colombo, F.

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, The Mathematics of Superoscillations (American Mathematical Society, 2017), Vol. 247.

Damborenea, J.

J. Damborenea, I. Egusquiza, G. Hegerfeldt, and J. Muga, “Measurement-based approach to quantum arrival times,” Phys. Rev. A 66, 052104 (2002).
[Crossref]

David, A.

de Abajo, F. J. G.

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical super-resolution through super-oscillations,” J. Opt. A 9, S285 (2007).
[Crossref]

De Lucia, F. C.

J. L. Gottfried, F. C. De Lucia, C. A. Munson, and A. W. Miziolek, “Laser-induced breakdown spectroscopy for detection of explosives residues: a review of recent advances, challenges, and future prospects,” Anal. Bioanal. Chem. 395, 283–300 (2009).
[Crossref]

Di Francia, G. T.

G. T. Di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento 9, 426–438 (1952).
[Crossref]

Dolev, S.

Dunin-Borkowski, R. E.

R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Superoscillating electron wave functions with subdiffraction spots,” Phys. Rev. A 95, 031802 (2017).
[Crossref]

Egusquiza, I.

J. Damborenea, I. Egusquiza, G. Hegerfeldt, and J. Muga, “Measurement-based approach to quantum arrival times,” Phys. Rev. A 66, 052104 (2002).
[Crossref]

Eleftheriades, G. V.

A. M. Wong and G. V. Eleftheriades, “An optical super-microscope for far-field, real-time imaging beyond the diffraction limit,” Sci. Rep. 3, 1715 (2013).
[Crossref]

Eliezer, Y.

Eveson, S. P.

S. P. Eveson, C. J. Fewster, and R. Verch, “Quantum inequalities in quantum mechanics,” in Annales Henri Poincaré (Springer, 2005), Vol. 6, pp. 1–30.

Fewster, C. J.

S. P. Eveson, C. J. Fewster, and R. Verch, “Quantum inequalities in quantum mechanics,” in Annales Henri Poincaré (Springer, 2005), Vol. 6, pp. 1–30.

Froim, S.

Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the temporal resolution limit by superoscillating optical beats,” Phys. Rev. Lett. 119, 043903 (2017).
[Crossref]

Gjonaj, B.

Gottfried, J. L.

J. L. Gottfried, F. C. De Lucia, C. A. Munson, and A. W. Miziolek, “Laser-induced breakdown spectroscopy for detection of explosives residues: a review of recent advances, challenges, and future prospects,” Anal. Bioanal. Chem. 395, 283–300 (2009).
[Crossref]

Greenfield, E.

Grübl, G.

M. Penz, G. Grübl, S. Kreidl, and P. Wagner, “A new approach to quantum backflow,” J. Phys. A 39, 423 (2005).
[Crossref]

Guaita, T.

F. Albarelli, T. Guaita, and M. G. Paris, “Quantum backflow effect and nonclassicality,” Int. J. Quantum Inf. 14, 1650032 (2016).
[Crossref]

Hadad, B.

Halliwell, J.

J. Yearsley and J. Halliwell, “An introduction to the quantum backflow effect,” in Journal of Physics: Conference Series (IOP Publishing, 2013), Vol. 442, p. 012055.

Hareli, L.

Y. Eliezer, B. K. Singh, L. Hareli, A. Arie, and A. Bahabad, “Experimental realization of structured super-oscillatory pulses,” Opt. Express 26, 4933–4941 (2018).
[Crossref]

Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the temporal resolution limit by superoscillating optical beats,” Phys. Rev. Lett. 119, 043903 (2017).
[Crossref]

Hegerfeldt, G.

J. Damborenea, I. Egusquiza, G. Hegerfeldt, and J. Muga, “Measurement-based approach to quantum arrival times,” Phys. Rev. A 66, 052104 (2002).
[Crossref]

Huang, F. M.

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical super-resolution through super-oscillations,” J. Opt. A 9, S285 (2007).
[Crossref]

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007).
[Crossref]

Hurwitz, I.

Javier Garcia de Abajo, F.

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007).
[Crossref]

Karimi, E.

Kreidl, S.

M. Penz, G. Grübl, S. Kreidl, and P. Wagner, “A new approach to quantum backflow,” J. Phys. A 39, 423 (2005).
[Crossref]

Leavens, C.

J. Muga, J. Palao, and C. Leavens, “Arrival time distributions and perfect absorption in classical and quantum mechanics,” Phys. Lett. A 253, 21–27 (1999).
[Crossref]

Leavens, C. R.

J. G. Muga and C. R. Leavens, “Arrival time in quantum mechanics,” Phys. Rep. 338, 353–438 (2000).
[Crossref]

Lechner, G.

H. Bostelmann, D. Cadamuro, and G. Lechner, “Quantum backflow and scattering,” Phys. Rev. A 96, 012112 (2017).
[Crossref]

Lobachinsky, L.

Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the temporal resolution limit by superoscillating optical beats,” Phys. Rev. Lett. 119, 043903 (2017).
[Crossref]

Longhi, S.

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
[Crossref]

Lu, P.-H.

R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Superoscillating electron wave functions with subdiffraction spots,” Phys. Rev. A 95, 031802 (2017).
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A. Bracken and G. Melloy, “Probability backflow and a new dimensionless quantum number,” J. Phys. A 27, 2197 (1994).
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J. L. Gottfried, F. C. De Lucia, C. A. Munson, and A. W. Miziolek, “Laser-induced breakdown spectroscopy for detection of explosives residues: a review of recent advances, challenges, and future prospects,” Anal. Bioanal. Chem. 395, 283–300 (2009).
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M. Palmero, E. Torrontegui, J. Muga, and M. Modugno, “Detecting quantum backflow by the density of a Bose-Einstein condensate,” Phys. Rev. A 87, 053618 (2013).
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M. Palmero, E. Torrontegui, J. Muga, and M. Modugno, “Detecting quantum backflow by the density of a Bose-Einstein condensate,” Phys. Rev. A 87, 053618 (2013).
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Muga, J. G.

J. G. Muga and C. R. Leavens, “Arrival time in quantum mechanics,” Phys. Rep. 338, 353–438 (2000).
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J. L. Gottfried, F. C. De Lucia, C. A. Munson, and A. W. Miziolek, “Laser-induced breakdown spectroscopy for detection of explosives residues: a review of recent advances, challenges, and future prospects,” Anal. Bioanal. Chem. 395, 283–300 (2009).
[Crossref]

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B. K. Singh, H. Nagar, Y. Roichman, and A. Arie, “Particle manipulation beyond the diffraction limit using structured super-oscillating light beams,” Light: Sci. Appl. 6, e17050 (2017).
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J. Muga, J. Palao, and C. Leavens, “Arrival time distributions and perfect absorption in classical and quantum mechanics,” Phys. Lett. A 253, 21–27 (1999).
[Crossref]

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M. Palmero, E. Torrontegui, J. Muga, and M. Modugno, “Detecting quantum backflow by the density of a Bose-Einstein condensate,” Phys. Rev. A 87, 053618 (2013).
[Crossref]

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F. Albarelli, T. Guaita, and M. G. Paris, “Quantum backflow effect and nonclassicality,” Int. J. Quantum Inf. 14, 1650032 (2016).
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M. Penz, G. Grübl, S. Kreidl, and P. Wagner, “A new approach to quantum backflow,” J. Phys. A 39, 423 (2005).
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D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
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Y. Aharonov, S. Popescu, and J. Tollaksen, “A time symmetric formulation of quantum mechanics,” Phys. Today 63 (11), 27 (2010).
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M. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A 39, 6965 (2006).
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Remez, R.

R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Superoscillating electron wave functions with subdiffraction spots,” Phys. Rev. A 95, 031802 (2017).
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R. Remez and A. Arie, “Super-narrow frequency conversion,” Optica 2, 472–475 (2015).
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G. Yuan, E. T. Rogers, and N. I. Zheludev, ““Plasmonics” in free space: observation of giant wavevectors, vortices, and energy backflow in superoscillatory optical fields,” Light: Sci. Appl. 8, 2 (2019).
[Crossref]

E. T. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15, 094008 (2013).
[Crossref]

Roichman, Y.

B. K. Singh, H. Nagar, Y. Roichman, and A. Arie, “Particle manipulation beyond the diffraction limit using structured super-oscillating light beams,” Light: Sci. Appl. 6, e17050 (2017).
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Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, The Mathematics of Superoscillations (American Mathematical Society, 2017), Vol. 247.

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Y. Eliezer, B. K. Singh, L. Hareli, A. Arie, and A. Bahabad, “Experimental realization of structured super-oscillatory pulses,” Opt. Express 26, 4933–4941 (2018).
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B. K. Singh, H. Nagar, Y. Roichman, and A. Arie, “Particle manipulation beyond the diffraction limit using structured super-oscillating light beams,” Light: Sci. Appl. 6, e17050 (2017).
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D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
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Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, The Mathematics of Superoscillations (American Mathematical Society, 2017), Vol. 247.

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B. Tamir and E. Cohen, “Introduction to weak measurements and weak values,” Quanta 2, 7–17 (2013).
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R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Superoscillating electron wave functions with subdiffraction spots,” Phys. Rev. A 95, 031802 (2017).
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Y. Aharonov, S. Popescu, and J. Tollaksen, “A time symmetric formulation of quantum mechanics,” Phys. Today 63 (11), 27 (2010).
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J. Tollaksen, “Novel relationships between superoscillations, weak values, and modular variables,” J. Phys. Conf. Ser. 70, 012016 (2007).
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Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, The Mathematics of Superoscillations (American Mathematical Society, 2017), Vol. 247.

Torrontegui, E.

M. Palmero, E. Torrontegui, J. Muga, and M. Modugno, “Detecting quantum backflow by the density of a Bose-Einstein condensate,” Phys. Rev. A 87, 053618 (2013).
[Crossref]

Tsur, Y.

R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Superoscillating electron wave functions with subdiffraction spots,” Phys. Rev. A 95, 031802 (2017).
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Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett 60, 1351 (1988).
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M. Penz, G. Grübl, S. Kreidl, and P. Wagner, “A new approach to quantum backflow,” J. Phys. A 39, 423 (2005).
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A. M. Wong and G. V. Eleftheriades, “An optical super-microscope for far-field, real-time imaging beyond the diffraction limit,” Sci. Rep. 3, 1715 (2013).
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J. Yearsley and J. Halliwell, “An introduction to the quantum backflow effect,” in Journal of Physics: Conference Series (IOP Publishing, 2013), Vol. 442, p. 012055.

Yuan, G.

G. Yuan, E. T. Rogers, and N. I. Zheludev, ““Plasmonics” in free space: observation of giant wavevectors, vortices, and energy backflow in superoscillatory optical fields,” Light: Sci. Appl. 8, 2 (2019).
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Zacharias, T.

Zheludev, N.

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007).
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Zheludev, N. I.

G. Yuan, E. T. Rogers, and N. I. Zheludev, ““Plasmonics” in free space: observation of giant wavevectors, vortices, and energy backflow in superoscillatory optical fields,” Light: Sci. Appl. 8, 2 (2019).
[Crossref]

E. T. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15, 094008 (2013).
[Crossref]

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical super-resolution through super-oscillations,” J. Opt. A 9, S285 (2007).
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ACS Photon. (1)

Y. Eliezer and A. Bahabad, “Super-oscillating airy pattern,” ACS Photon. 3, 1053–1059 (2016).
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Anal. Bioanal. Chem. (1)

J. L. Gottfried, F. C. De Lucia, C. A. Munson, and A. W. Miziolek, “Laser-induced breakdown spectroscopy for detection of explosives residues: a review of recent advances, challenges, and future prospects,” Anal. Bioanal. Chem. 395, 283–300 (2009).
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G. Allcock, “The time of arrival in quantum mechanics III. The measurement ensemble,” Ann. Phys. (N. Y.) 53, 311–348 (1969).
[Crossref]

Appl. Phys. Lett. (1)

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007).
[Crossref]

Bell Syst. Tech. J. (2)

S. A. Schelkunoff, “A mathematical theory of linear arrays,” Bell Syst. Tech. J. 22, 80–107 (1943).
[Crossref]

D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[Crossref]

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M. Berry, “Five momenta,” Eur. J. Phys. 34, 1337 (2013).
[Crossref]

Int. J. Quantum Inf. (1)

F. Albarelli, T. Guaita, and M. G. Paris, “Quantum backflow effect and nonclassicality,” Int. J. Quantum Inf. 14, 1650032 (2016).
[Crossref]

J. Opt. (1)

E. T. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15, 094008 (2013).
[Crossref]

J. Opt. A (1)

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical super-resolution through super-oscillations,” J. Opt. A 9, S285 (2007).
[Crossref]

J. Phys. A (5)

A. Bracken and G. Melloy, “Probability backflow and a new dimensionless quantum number,” J. Phys. A 27, 2197 (1994).
[Crossref]

M. Penz, G. Grübl, S. Kreidl, and P. Wagner, “A new approach to quantum backflow,” J. Phys. A 39, 423 (2005).
[Crossref]

M. Berry, “Quantum backflow, negative kinetic energy, and optical retro-propagation,” J. Phys. A 43, 415302 (2010).
[Crossref]

M. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A 39, 6965 (2006).
[Crossref]

M. Berry and P. Shukla, “Pointer supershifts and superoscillations in weak measurements,” J. Phys. A 45, 015301 (2011).
[Crossref]

J. Phys. Conf. Ser. (1)

J. Tollaksen, “Novel relationships between superoscillations, weak values, and modular variables,” J. Phys. Conf. Ser. 70, 012016 (2007).
[Crossref]

Laser Photon. Rev. (1)

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
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Light: Sci. Appl. (2)

G. Yuan, E. T. Rogers, and N. I. Zheludev, ““Plasmonics” in free space: observation of giant wavevectors, vortices, and energy backflow in superoscillatory optical fields,” Light: Sci. Appl. 8, 2 (2019).
[Crossref]

B. K. Singh, H. Nagar, Y. Roichman, and A. Arie, “Particle manipulation beyond the diffraction limit using structured super-oscillating light beams,” Light: Sci. Appl. 6, e17050 (2017).
[Crossref]

Nuovo Cimento (1)

G. T. Di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento 9, 426–438 (1952).
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Opt. Express (2)

Opt. Lett. (2)

Optica (3)

Photon. Spectra (1)

V. C. Coffey, “Advances in standoff detection make the world safer,” Photon. Spectra 47, 44–47 (2013).

Phys. Lett. A (1)

J. Muga, J. Palao, and C. Leavens, “Arrival time distributions and perfect absorption in classical and quantum mechanics,” Phys. Lett. A 253, 21–27 (1999).
[Crossref]

Phys. Rep. (1)

J. G. Muga and C. R. Leavens, “Arrival time in quantum mechanics,” Phys. Rep. 338, 353–438 (2000).
[Crossref]

Phys. Rev. A (4)

J. Damborenea, I. Egusquiza, G. Hegerfeldt, and J. Muga, “Measurement-based approach to quantum arrival times,” Phys. Rev. A 66, 052104 (2002).
[Crossref]

R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Superoscillating electron wave functions with subdiffraction spots,” Phys. Rev. A 95, 031802 (2017).
[Crossref]

M. Palmero, E. Torrontegui, J. Muga, and M. Modugno, “Detecting quantum backflow by the density of a Bose-Einstein condensate,” Phys. Rev. A 87, 053618 (2013).
[Crossref]

H. Bostelmann, D. Cadamuro, and G. Lechner, “Quantum backflow and scattering,” Phys. Rev. A 96, 012112 (2017).
[Crossref]

Phys. Rev. Lett (1)

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett 60, 1351 (1988).
[Crossref]

Phys. Rev. Lett. (1)

Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the temporal resolution limit by superoscillating optical beats,” Phys. Rev. Lett. 119, 043903 (2017).
[Crossref]

Phys. Today (1)

Y. Aharonov, S. Popescu, and J. Tollaksen, “A time symmetric formulation of quantum mechanics,” Phys. Today 63 (11), 27 (2010).
[Crossref]

Quanta (1)

B. Tamir and E. Cohen, “Introduction to weak measurements and weak values,” Quanta 2, 7–17 (2013).
[Crossref]

Sci. Rep. (1)

A. M. Wong and G. V. Eleftheriades, “An optical super-microscope for far-field, real-time imaging beyond the diffraction limit,” Sci. Rep. 3, 1715 (2013).
[Crossref]

Other (4)

M. Berry, Quantum Coherence and Reality: In Celebration of the 60th Birthday of Yakir Aharonov (World Scientific, 1994).

J. Yearsley and J. Halliwell, “An introduction to the quantum backflow effect,” in Journal of Physics: Conference Series (IOP Publishing, 2013), Vol. 442, p. 012055.

S. P. Eveson, C. J. Fewster, and R. Verch, “Quantum inequalities in quantum mechanics,” in Annales Henri Poincaré (Springer, 2005), Vol. 6, pp. 1–30.

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, The Mathematics of Superoscillations (American Mathematical Society, 2017), Vol. 247.

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Figures (4)

Fig. 1.
Fig. 1. Finite backflow function $ {f_{{\rm FBF}}}(\xi ) $. (Left) Backflow function spatial spectrum. $ {k_0} $ is the fundamental spatial frequency. The dashed black line represents the center of the $ k $ axis, related to zero transverse momentum. (Center) Backflow function in real space. (Right) Local spatial frequency of $ {f_{{\rm FBF}}}(\xi ) $. The rows correspond to (a) $ a = 1.0 $, (b) $ a = 0.7 $, and (c) $ a = 0.4 $.
Fig. 2.
Fig. 2. Experimental setup. BE, beam expander; SLM, spatial light modulator; MS, moving stage; SL, slit; M, mirror; $ {L_1},{L_2},{L_3},{L_4} $, lenses. $ {d_1} + {d_2} $ equals lens $ {L_3} $ focal length. $ {Z_1} $ and $ {Z_2} $ mark the locations of the first and second focal planes, respectively. (Inset) (I) Realization of one of the phase-only masks used in the experiment. (II) Corresponding intensity of the beam at the first diffraction order at the first focal plane.
Fig. 3.
Fig. 3. Experimental measurements. (Left) Generated SLM phase-only masks. Each line creates a propagating mode with a well-defined negative transverse momentum. The dotted line represents the center of the $ x $ axis, related to zero transverse momentum. (Center) Measured intensity distribution (in counts) in the first focal plane (backflow beam). The two dashed white lines represent the width of the slit. (Right) Measured beam image at the second focal plane, averaged over the $ y $ coordinate for each slit position. The dashed-dotted black line denotes the center of the propagation axis. Continuous red line: measured expectation value of the beam position (equal to momentum in the first focal plane). Dashed blue line: analytically calculated expectation value for a theoretical infinite periodic backflow beam after it is slit-filtered. Dotted green line: expectation value derived from Fourier transforming the SLM image and then Fourier-transforming again the slit-filtered image. (a) $ a = 1 $, (b) $ a = 0.7 $, and (c) $ a = 0.4 $.
Fig. 4.
Fig. 4. Intensity distribution for different slit widths. Simulated beam image at the second focal plane, averaged over the $ y $ coordinate for each slit position, for the case of $ a = 0.4 $ and for different values of the slit’s width: (left) $ W = 50\;{\unicode{x00B5}{\rm m}} $, (center) 100 µm, and (right) 200 µm. The dotted green curve represents the expectation value of the beam's position.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

f B F ( ξ ) = f S u b ( ξ ) exp ( i L ξ ) = exp ( i L ξ ) [ cos ( k 0 ξ ) + i a sin ( k 0 ξ ) ] N ,
F B F ( k ) = 2 π m = + C m ( a ) δ ( k L m k 0 ) ,
k l o c a l ( ξ ) = I m ln [ f B F ( ξ ) ] ξ = L N a k 0 cos 2 ( k 0 ξ ) + a 2 sin 2 ( k 0 ξ ) .
C m ( a ) = { ( m 2 1 ) ( a + 1 ) m 2 3 2 ( a 1 ) m 2 + 3 2 , m { o d d < 0 } , 0 , o t h e r w i s e .
F F B F ( k ) = m = P N C m ( a ) exp ( [ k L m k 0 ] 2 2 σ 0 2 ) ,
k l o c a l ( ξ ) = I m ln [ ψ ( ξ ) ] ξ = R e ξ | k ^ | ψ ξ | ψ = A w e a k .

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